Result Details
Alexandroff pretopologies for structuring the digital plane
We explore the possibility of employing Alexandroff pretopologies as structures on the digital plane Z^2 convenient for the study of geometric and topological properties of digital images. These pretopologies are known to be in one-to-one correspondence with reflexive binary relations so that graph-theoretic methods
may be used when investigating them. We discuss such Alexandroff pretopologies on Z2 that possess a rich variety of digital Jordan curves obtained as circuits in a natural graph with the vertex set Z2. Of these pretopologies, we focus on the minimal ones and study their quotient pretopologies on Z2 which are shown
to allow for various digital Jordan curve theorems. We also develop a method for identifying Jordan curves in the minimal pretopological spaces by using Jordan curves in one of their quotient spaces. Using this method, we conclude the paper with proving a digital Jordan curve theorem for the minimal pretopologies.
Digital plane, Jordan curve, Alexandroff pretopology, quotient pretopology
@article{BUT125480,
author="Josef {Šlapal}",
title="Alexandroff pretopologies for structuring the digital plane",
journal="DISCRETE APPLIED MATHEMATICS",
year="2017",
volume="216",
number="2",
pages="323--334",
doi="10.1016/j.dam.2016.06.002",
issn="0166-218X",
url="https://ac.els-cdn.com/S0166218X16302670/1-s2.0-S0166218X16302670-main.pdf?_tid=b5db0aee-e1e1-11e7-b51a-00000aab0f02&acdnat=1513374708_82e3d74b75420ea8adca800b18dc4e43"
}